Figure 20 illustrates flow pattern in the laminar flow region from a radial flat blade turbine. By using a velocity probe, the parabolic velocity distribution coming off the blades of the impeller is shown in Fig. 21. By taking the slope of the curve at any point, the shear rate may be calculated at that point. The maximum shear rate around the impeller periphery as well as the average shear rate around the impeller may also be calculated.

An important concept is that one must multiply the fluid shear rate from the impeller by the viscosity of the fluid to get the fluid shear stress that actually carries out the process of mixing and dispersion.

Fluid shear stress = n(fluid shear rate)

Even in low viscosity fluids, by going from 1 cp to 10 cp there will be 10 times the shear stress of the process operating from the fluid shear rate of the impeller.

Figu re 20. Photograph of radial flow impeller in a baffled tank in the laminar region, made by passing a thin plane of light through the center of the tank.

Figu re 21. Typical velocity pattern coming from the blades of a radial flow turbine showing calculation of the shear rate AF/A7.

Figure 22 shows the flow pattern when there is sufficient power and low enough viscosity for turbulence to form. Now a velocity probe must be used that can pick up the high frequency response of these turbulent flow patterns, and a chart as shown in Fig. 23 is typical. The shear rate between the small scale velocity fluctuations is called microscale shear rate, while the shear rates between the average velocity at this point are called the macroscale rates. These macroscale shear rates still have the same general form and are determined the same way as shown in Fig. 21.

Figure 22 shows the flow pattern when there is sufficient power and low enough viscosity for turbulence to form. Now a velocity probe must be used that can pick up the high frequency response of these turbulent flow patterns, and a chart as shown in Fig. 23 is typical. The shear rate between the small scale velocity fluctuations is called microscale shear rate, while the shear rates between the average velocity at this point are called the macroscale rates. These macroscale shear rates still have the same general form and are determined the same way as shown in Fig. 21.

Figure 22. Photograph of flow patterns in a mixing tank in the turbulent region, made by passing a thin plane of light through the center of the tank.

Table 2 describes four different macroscale shear rates of importance in a mixing tank. The parameter for the microscale shear rate at a point is the root mean square velocity fluctuation at that point, RMS.

Table 2. Average Point Velocity

Max. imp. zone shear rate Ave. imp. zone shear rate Ave. tank zone shear rate Min. tank zone shear rate

RMS velocity fluctuations

5.1 Particles

The consideration of the macro- and microscale relationships in a mixing vessel leads to several helpful concepts. Particles that are greater 1,000 microns in size are affected primarily by the shear rate between the average velocities in the process and are an essential part of the overall flow throughout the tank and determine the rate at which flow and velocity distribute throughout the tank, and is a measure of the visual appearance of the tank in terms of surface action, blending or particle suspensions.

The other situation is on the microscale particles. They are particles less than 100 microns and they see largely the energy dissipation which occurs through the mechanism of viscous shear rates and shear stresses and ultimately the scale at which all energy is transformed into heat.

The macroscale environment is effected by every geometric variable and dimension and is a key parameter for successful scaleup of any process, whether microscale mixing is involved or not. This has some unfortunate consequences on scaleup since geometric similarity causes many other parameters to change in unusual ways, which may be either beneficial or detrimental, but are quite different than exist in a smaller pilot plant unit. On the other hand, the microscale mixing condition is primarily a function of power per unit volume and the result is dissipation of that energy down through the microscale and onto the level of the smallest eddies that can be identified as belonging to the mixing flow pattern. An analysis of the energy dissipation can be made in obtaining the kinetic energy of turbulence by putting the resultant velocity from the laser velocimeter through a spectrum analyzer. Figure 24a shows the breakdown of the energy as a function of frequency for the velocities themselves. Figure 24b shows a similar spectrum analysis of the energy dissipation based on velocity squared and Fig. 24c shows a spectrum analyzer result from the product of two orthogonal velocities, VR and Vz' which is called the Reynolds stress (a function of momentum).

An estimation method of solving complex equations for turbulent flow uses a method called the K-s technique which allows the solution of the Navier-Stokes equation in the turbulent region.

Figure 25 shows a typical Reynolds number-Power number curve for different impellers. The important thing about this curve is that it holds true whether the desired process job is being done or not. Power equations have three independent variables along with fluid properties: power, speed and diameter. There are only two independent choices for process considerations.

For gas-liquid operations there is another relationship called the K factor which relates the effect of gas rate on power level. Figure 26 illustrates a typical K factor plot which can be used for estimation. Actual calculation of AT factor in a particular case involves very specific combinations of mixer variables, tank variables, and fluid properties, as well as the gas rate being used.

Commonly, a physical picture of gas dispersion is used to describe the degree of mixing required in an aerobic fermenter. This can be helpful on occasion, but often gives a different perspective on the effect of power, speed and diameter on mass transfer steps. To illustrate the difference between physical dispersion and mass transfer, Fig. 27 illustrates a measurement made in one experiment where the height of a geyser coming off the top of the tank was measured as a function of power for various impellers. Reducing the geyser height to zero gives a uniform visual dispersion of gas across the surface of the tank. Figure 28 shows the actual data and indicates that the 8-inch impeller was more effective than the 6-inch impeller in this particular tank.

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